'^^/ ; LIB R.AFLY OF THE U N IVER.SITY OF ILLI NOIS 621.365 I^G55te no. 40-49 cop. 2 Digitized by the Internet Archive in 2013 http://archive.org/details/newcircularlypol46dyso ANTENNA LABORATORY Technical Report No. 46 NEW CIRCULARLY POLARIZED FREQUENCY INDEPENDENT ANTENNAS WITH CONICAL BEAM OR OMNIDIRECTIONAL PATTERNS by John D. Dyson and Paul E. Mayes 20 June 1960 Contract AF33(616)-6079 Project No. 9-(13-6278) Task 40572 Sponsored by: WRIGHT AIR DEVELOPMENT CENTER Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois This report was presented, in condensed form, as a paper at the URSI-IRE Spring Meeting, Washington, D.C., 3 May 1960, under the title, "The Log-Spiral Omnidirectional Circularly Polarized Antenna." i i 3 \pZ> - ENGINEERING LIBRARY ACKNOWLEDGMENT The authors are pleased to acknowledge the assistance of Professor G. A. Deschamps in formulating the theory of excitation of multi-arm antennas. Discussions with W. T. Patton were helpful. 0. L. McClelland supervised the measurements program. ABSTRACT A conical beam may be obtained from balanced equiangular spiral antennas by constructing an antenna with more than two spiral arms and symmetrically connecting these arms to provide a suppression of the radiated fields on the axis of the antenna. The angle of this conical beam can be controlled and with proper choice of parameters it can be confined to the immediate vicinity of the azimuthal (0 = 90 ) plane. An antenna with four symmetrically spaced arms can provide a radiation pattern that is within 3 db of omnidirectional circularly polarized coverage. The standing wave ratio of this antenna referred to a 50 ohm coaxial cable is less than 2 to 1 over the pattern bandwidth, This four-arm version retains the wide frequency bandwidths of the basic conical log-spiral antenna, and it provides a coverage which here- tofore has been difficult to obtain even with narrow band antennas. CONTENTS Page 1. Introduction 1 2. The Conical Log-Spiral Antenna 3 2.1 The Basic Structure 3 2.2 The Radiation Pattern Beamwidth 5 3. The Conical Beam Antenna 7 3.1 The Principle 7 3.2 Radiation Patterns 9 3.3 Pattern Beamwidth 17 3.4 The Input Impedance 17 3.5 Operating Bandwidth as a Function of Antenna Size 17 4. A Non-Frequency Independent Version 20 5. Conclusions 24 Appendix 25 ILLUSTRATIONS Figure Number Page 1. A conical log-spiral antenna with associated coordinate system 4 2. Variation in electric field pattern of typical balanced 2 arm conical equiangular spiral antenna 6 3. Possible feeding arrangements for multi-arm structures 8 4. "infinite balun" feed used on a four arm conical beam antenna 10 5. Typical electric field patterns and orientation of the conical beam as a function of the rate of spiral 11 6. Antenna C - 15 - 9 etched from copper clad teflon impregnated fiberglass 13 7. Electric field patterns of a balanced symmetrical 4 arm conical equiangular spiral antenna 14 8. Electric field patterns of a balanced symmetrical 4 arm conical equiangular spiral antenna 15 9. Azimuthal coverage of the radiation patterns in Figures 7 & 8 16 10. VSWR of typical 4 arm conical equiangular spiral 18 11. Projection of equiangular spiral and Archimedes spiral curves on a conical surface 21 12. Electric field patterns of symmetrical 4 arm conical antennas 23 1A. Terminal region of a structure having N-fold rotational symmetry 26 2A. Instataneous electric vectors at 9 = and 6=9 for three values of m ° 30 1. INTRODUCTION The balanced planar and conical equiangular spiral antennas have been demonstrated to have essentially frequency independent radiation and impedance characteristics over bandwidths which are at the discretion of the design 12 * engineer ' . These antennas, based upon the equiangular or log-spiral curve, have the property that the highest and lowest usable frequencies are independent The highest usable frequency is determined by the diameter of the truncated region at the origin, which must remain small in terms of the operating wavelength, and the lowest usable frequency by the arm length and hence the maximum diameter of the antenna. The two-arm planar antenna provides circularly polarized, single lobe, bidirectional radiation on the axis of the antenna. An orthogonal projection of the two-arm planar antenna on a conical surface forms an antenna which, over a suitable range of parameters, confines the radiation to a single lobe directed off the apex of the cone. It is possible to devise a multitude of frequency independent structures by using the log-spiral arm as a basic element. Many different excitations may be used when several log-spiral arms with a common origin are placed on a cone or plane. Some configurations and excitations produce radiation patterns which are distinctly different from those obtained heretofore. * A convenient abbreviation for logarithmic spiral, a synonym for equiangular spiral. 1. J. D. Dyson, "The Equiangular Spiral Antenna," IRE Trans, on Antennas and Propagation, vol. AP-7, pp. 181-187, April, 1959. Also Technical Report No. 21, University of Illinois Antenna Laboratory, September 15, 1957. 2. J. D. Dyson, "The Unidirectional Equiangular Spiral Antenna," IRE Trans.. on Antennas and Propagation, Vol. AP-7, October 1959* Also Technical Report No. 33, University of Illinois, Antenna Laboratory, July 10, 1958. It is the purpose of this paper to introduce a simple theory relating the excitation and the radiation fields (insofar as now possible), and to present data showing the performance of the four-arm conical log-spiral antenna with one particular excitation. 2. THE CONICAL LOG-SPIRAL ANTENNA 2.1 The Basic Structure The conical log-spiral antenna with its associated coordinate system is shown in Figure 1. On a plane surface, the edges of one arm of a logarithmic spiral antenna may be defined by '■ ' i aff P = P e and / a a(<P - $) Since Tan a = dPV.d0> . a / / ( — - — ) <P P l =P e tan a The orthogonal projection of P and P_ on the surface of revolution 1 ^ 7T - 9 = G is defined by sin 9 tan a ' r sin f a " d -• (r -2) (<P-h where • -sin 6 2_ 6 . / tan a K = e P is the radius vector from the origin to the truncation of the spiral at the apex region, and P and p are the radius vectors to the inner and outer edges of the exponentially expanding arm at a given angle <P. The angle ° is a constant and of such value that if the curve traced out by P is rotated about the axis through the angle 6 it will coincide with the curve traced out by p . The angle a, a constant, is the angle between the radius vector and a tangent to the log-spiral curve at the point of inter- section. Figur* 1 A conical log-spiral antenna with associated coordinate system 5 The second arm of the balanced structure is defined by rotating curves 1 and 2 through 7T radians. Hence the defining parameters are: the included cone angle, 26 ; the arm width determined by °, or K; the rate of spiral, a, the base diameter, D; and the apex diameter, d. 2.2 The Radiation Pattern Beamwidth The beam width of the two-arm conical antenna can be controlled over a limited range by a suitable choice in the rate of spiral. Figure 2 shows typical electric field radiation patterns for balanced two arm .o „„o , ,_o ant ennas constructed with an a of 73 60 and 45 . Typical half-power beamwidths range from 60-70° for a of 82 , 70-80 for a of 73°, and 160-180° o o for an a of 60 . As the angle a is decreased to 45 the beamwidth increases o to 180-200 . Pattern cuts through the axis and perpendicular to the axis of the antenna [Figure 2(c)] indicate that this latter case provides essentially circularly polarized coverage in one hemisphere and omnidirectional coverage on the 6 = 90 plane. It has been pointed out that a modified version of the balanced conical antenna is obtained when the width of the arm is made constant rather than 2 tapered . This form of the antenna is readily constructed of wire or cable. However, it is only an approximation to the true equiangular spiral structure. The approximation is good for relatively tightly spiraled antennas, i.e., a greater than 60 , and results in only minor pattern changes. As the angle a is decreased to the neighborhood of 45 to 50 marked pattern changes occur for the wire approximation, including a multilobing of the main beam and large radiation off the base of the cone. Thus to realize the patterns such as shown in Figure 2(c) the decrease in the rate of spiral must be accompanied by an increase in the arm width, i.e. increase in o (or K) . The patterns shown o in Figure 2(c) were for an antenna constructed with the parameters; 29 = 20 , o a = 45° K = .75 ( 6 = 94°). <f> =0°, 9 VAR (1=73' <£VAR, e=90 < ^.=60* b. «=45 ( Figure 2 Variation in electric field pattern of typical balanced 2 arm conical equiangular spiral antenna E^, Eg polarization. 9 = 10 . o 3. THE CONICAL BEAM ANTENNA 3.1 The Principle When using multiple-arm structures the number of choices of feeding 3 systems increases . There are basic excitations of multiterminal antennas which are simply related to the azimuthal variations of fields of the - -jirfp form e J associated with solutions of Maxwell's equations. The parameter m must be an integer to make the field single-valued. Excitations of the spiral arms which correspond to each of these radiation "modes" are readily apparent. The customary excitation of the two-arm spirals, as shown in Figure 3(a), corresponds to m = + 1 and would be expected to produce a field which varies primarily as e . For antennas which are not large compared to the wavelength the lower order terms will be predominant. With four-arm structures the number of basic excitations increases. An excitation corresponding to m = 1 is shown in Figure 3(b) . Hence in order to obtain the e fields with a four-arm spiral it is necessary to excite the two pairs of arms with a 90 degree phase shift between them. With the four-arm structures it is possible to produce fields which correspond to higher values of m . For example, the excitation shown in Figure 3(c) corresponds to m = +2 and should produce fields which vary -i2fP primarily as e . This concept may be generalized to any number of arms, N, and a discussion of the more general case is given in the Appendix. 3. G. A. Deschamps, "impedance Properties of Complementary Multiterminal Planar Structures," Trans. IRE, Special Supplement, Vol. AP-7, Dec, 1959, p. S371. Also Technical Report No. 43, University of Illinois, Antenna Laboratory, Nov. 11, 1959. * First pointed out in Quarterly Report No. 5, Contract AF33(616)-6079, Antenna Laboratory, University of Illinois, 31 December 1959, pp. 9-11. + CT7 (a) +j -I + IC=3 (b) "J fl D fl Figure 3 Possible feeding arrangements for multi-arm structures 9 Examination of the solutions of Maxwell's equations in spherical coordinates shows that values of m different from unity are always accompanied by a null in the 9 - function (associated Legendre polynomials) along the polar or 8 = axis. Therefore we expect an excitation which corresponds to any in / + 1 to produce a conical beam. The excitation of four arms corresponding to m = 2 as shown in Figure 3(c) , is the simplest case. This lowest order conical beam excitation is readily achieved by connecting opposite arms together and feeding one pair against the other, i.e. 180 degrees out of phase. It is apparent also from the symmetry of the input currents in this case that there will be zero field along the antenna axis. The antenna can be fed by a balanced feed line, or a coaxial line and balun, placed on the axis of symmetry. It may also be fed by carrying the feed cable along one of the arms as outlined in the previous 2 paper m Details of this latter method are shown in Figure 4. The balance and symmetry of the feed is important if symmetrical patterns are desired. 3.2 Radiation Patterns Figure 5 shows typical radiation patterns of symmetrical four arm antennas fed in the manner shown in Figure 3(c). As indicated, the rate of spiral, (the parameter a), which was the primary factor in controlling the beamwidth of the balanced two arm antenna, determines the orientation of the conical beam of the balanced four arm antenna. Conical antennas may be constructed to provide a conical beam with any angle of orientation from around 40 to more than 90 off the axis of the antenna. The case where the beam maximum is located at 9 = 90 is of particular interest since it fills a need for a simple, very broad band, circularly polarized, omnidirectional source. 10 \ \ TOP \ V SIDE Figure 4 'infinite balun" feed used on a four arm conical beam antenna <*=73° oC = 60° 11 Q^=45° uj UJ o CD 40 r 50 - 60 70 - - 80 90 /' •J- 40 50 60 70 o< IN DEGREES 80 90 Figure 5 Typical electric field patterns and orientation of the conical beam as a function of the rate of spiral (7.5 ^ < 10 ) • ■ . Figure 6 shows a typical four arm balanced equiangular spiral antenna constructed on a 15 cone. This antenna was etched from a flexible, copper- clad, teflon-impregnated, fiberglass material and then formed into a cone. The feed cable is rg 141/U. The energized cable is carried along one arm; dummy cables are placed on the other arms to maintain structural symmetry. To obtain the desired bandwidth, the arms on this particular antenna were later extended to a cone base diameter of 31 centimeters. Radiation patterns of this antenna are shown in Figures 7 and 8 from 550 mc where the base is .57 wavelengths in diameter up to 4000 mc where the diameter of truncated apex is approximately 0.2 wavelengths. The patterns are for E~ and E.~ polarized fields. The first two columns are pattern cuts through the axis of the antenna and the third column is for a cut perpendicular to the axis, on the U = 90 plane. The azimuthal coverage shown in these patterns may be examined in greater detail in Figure 9 where the total deviation in decibels from omnidirectional coverage is plotted for the orthogonal polarizations. The axial ratio on the 6 = 90 plane varies somewhat with the angle i P. It is shown for one particular angle, which is a representative angle of orientation and not an optimum case. Over a considerable bandwidth the total amplitude deviation is less than 3 db and the axial ratio is 3 db or less. Two pattern characteristics should be noted. For large a, the conical beam patterns are smooth and well formed and, if desired, the arms may be approximated by wire or cable. As the angle a decreases beyond 60° the on patterns exhibit minor irregularities and are not as symmetrical. In ;i ■« loosely spiraled antennas require wider exponentially expanding arms. Th< ••-.<■ ' h. istics correspond to those noted for the 13 Figure 6 Antenna C - 15 - 9 etched from copper clad teflon impregnated fiberglass a = 45° D ■ 20.5 cm 14 8°, OVcuu. F=550Mc 700 Mc CJ>-90° e vcuo. (j)Vojo Q = 9Q < 900 Mc N \ S \ / r ^ 1200 Mc Figure 7 Electric field patterns of a balanced symmetrical 4 arm conical ilangular spiral antenna. .", a 1 .", K = .925, D = 31 cm, d = 1.5 cm 0s 0° e "z^t/. 1600 Mc ^=90° e vcuu. (£ VoMj. ^ Q r 9QO 15 3000 Mc ^ "h 'y.c. .^'^^s-:^ V-< ' -v 7 ^?r 4000 Mc y s; ^r=— \ n I : \ ' / I / -^~ e = Figure 8 Electric field patterns of a balanced symmetrical 4 arm conical equiangular spiral antenna. 7.5°, a = 45°, K = .925, D = 31 cm, d = 1. 5 cm a o •H +-> cS •H •o <D X! 00 -P eg «H O «> ID (0 bD <D Q) U> > 'H O Ek o c •H (0 c Fh 0) -p +-> a a X! ■p S •H N a) h •H o08l = 4> o06=e OllVti 1VIXV qp Nl NfcGlJMd Nonviavd o06=e jo N0I1VIA3Q 3anilldkNV 17 two arm axial beam antennas. 3.3 Pattern Beamwidth The beamwidth in a <P = constant plane is relatively insensitive to a change in antenna parameters. Antennas constructed with both 15 and 20 included cone angles (2« ) with 45 < a < 73 and with cable arms or with o - - exponentially expanding arms had half-power beamwidths ranging from 35 to 55 degrees, with an average value of 45 degrees. 3.4 The Input Impedance The input impedance of the four arm antenna, fed in the manner of Figure 3(c), rapidly converges to a characteristic value. Antennas constructed o o with 15 or 20 included cone angles with RG 8/U arms or exponentially expanding arms fed with RG 141/U, typically have an input impedance of from 45 to 55 ohms for a ranging from 45 to 60 degrees. As a is increased to 73 degrees the impedance rises to the neighborhood of 70 ohms. These values are approximately one half those noted for similar two arm antennas. The input voltage standing wave ratio of the antenna referred to in Figures 7 and 8 is plotted in Figure 10. Note that it is less than 1.5 to 1 referred to 50 ohms over most of the usable pattern bandwidth. 3.5 Operating Bandwidth as a Function of Antenna Size The usable antenna bandwidth is fundamentally determined by the diameter of the truncated apex and the antenna arm length. As with the two arm antennas, the radiation patterns tend to deteriorate as the apex region approaches 1/4 wavelength in diameter. It was previously noted that the balanced two arm antenna constructed on a 15 or 20 degree (total apex angle) cone, with an a of 73 could be operated to a frequency such that the cone base diameter is on the order of 1/3 wavelength. As the rate of spiral ed U •H C o o E u a •<# u •H h a O •H U a eS >> rH tuD <H C rt •H ^ cr to <D > s-t bO « e o ro CD in <* ll U~ OS 01 Q3dd3J3d tiMSA 19 is decreased, i.e. a decreased to 45 there is not sufficient radiation surface on this size cone to dissipate the energy without back radiation and hence the size of the cone must be increased to the order of .6 wavelength at the lowest frequency of operation. The four arm structures exhibit very similar characteristics and hence omnidirectional coverage on the = 90 plane requires an antenna whose base diameter is on the order of .6 to 2/3 wavelength at the lowest operating frequency. 20 4. A NON-FREQUENCY INDEPENDENT VERSION Thus far we have considered only antennas constructed from the equiangular spiral curve. These antennas are frequency independent in the sense that, within the limits imposed by the physical size, the 4 scaling principle is fulfilled . The pattern characteristics of these log-spiral antennas (such as the beamwidth of the two arm antennas and the angle of orientation of the conical beam of the four arm antennas) are constant for a change in the frequency of operation. These characteristics are directly related to the constant parameter a, which indicates the rate of spiraling of the arms. It is possible, as shown in Figure 11, to construct conical antennas from other curves, such as the Archimedes spiral. Although these antennas may be operated over wide frequency bands, they are not frequency independent since the parameter a at any point on the curve is directly related to the angle <P at that point. As the frequency of operation is changed the active aperture of the antenna is composed of a structure with a changing rate of spiral. This shows up as a definite widening of the beamwidth of the two arm conical Archimedes spiral antenna as the operating frequency is increased. There is also a variation in the angle of orientation of the conical beam o! the four arm conical Archimedes spiral antenna with a change in frequency. A. V. H. Kurmey, Frequency Independent Antennas," 1957, IRE National "1, P'. 1, pp. 114-118. Also Technical Report No. 20, University '-f tlllnoie, Antenna Laboratory, October 25, 1957. p =ke Q(j> 21 a = arctan tt (a) p = ka<£ << = arctan <£ (b) Figure 11 Projection of equiangular spiral and Archimedes spiral curves on a conical surface. 22 Radiation patterns for a four arm conical Archimedes spiral antenna are shown in Figure 12(a) . This antenna was constructed to provide a range o o of a from approximately 45 at the apex region to 85 at the base. As indicated in Figure 12 the complete range of beam orientation from o o approximately 45 to 90 off axis is swept out as the frequency is varied from 1000 to 2000 mc . For comparison, patterns for an equiangular spiral antenna wound on the same cone are shown in Figure 12(b) . ARCHIMEDES SPIRAL P =1.026* f = 1000 Mc E * B EQUIANGULAR SPIRAL ( />= e sin 10° tan45° )4> 23 f = 1400 Mc f = 2000 Mc Figure 12 Electric field patterns of symmetrical 4 arm conical antennas © = 10°, D = 29.5 cm, d = 4.5 cm o ' ' (<P = 90°, Q var pattern) 24 5. CONCLUSIONS It is possible to obtain a conical beam from the balanced equiangular spiral antenna by constructing the antenna with more than two arms and connecting these arms to provide a higher order <P - variation which is accompanied by an axial null. The angle of this conical beam can be controlled; in particular, it can be placed in the immediate o vicinity of the 6 = 90 plane to provide an omnidirectional pattern. An antenna with four symmetrically spaced arms can provide a radiation pattern that is within 3 db of omnidirectional coverage. The standing wave ratio of this antenna, referred to a 50 ohm coaxial cable, is usually less than 1.5 to 1 over the pattern bandwidth. This form of log-spiral retains the extremely wide frequency band- width and circular polarization properties of the basic conical log- spiral antenna and it provides a coverage which heretofore has been difficult to obtain even with narrow band antennas. Since the Archimedian spiral approximates a log-spiral with changing parameters, this non-frequency-independent version of the antenna may be constructed such that the conical beam may be frequency scanned in the polar angle. 25 APPENDIX Consider a radiating system composed of a number of identical conductors on the surface of a cone. Assume that the structure has N-fold rotational 2ir symmetry, i.e., a rotation about the cone axis through the angle will leave the structure unchanged. Figures 3(b) and 3(c) of the text show examples of four-fold symmetry. The excitation of the conductors will be accomplished at the apex of the cone where the dimensions are small compared to the wavelength. There will be N terminals available in the excitation region, symmetrically spaced about a circle of small radius. Denoting the input current at the n terminal by I , the excitation can be described by the current vector I = ( V V v v w Any index is defined modulo N. To satisfy the requirement of conservation of current, we note that N £ 1=0 (1A) T n n = 1 This viewpoint that each possible excitation is represented by a vector, I leads to the examination of the possible basis for the vectors of this space. Because of the n-fold symmetry the choice of "symmetrical components" as base vectors proves more convenient than others. Rotation of the excitation by one step : a = ( v v v V (2A) 26 ® / © © © © Figure 1A Terminal region of a structure having N - fold rotational symmetry 27 2 IT would simply produce a field which is rotated in space by the angle " Hence the transformation of excitation I 1 as PI (3A) where P is th*~ n x n permutation matrix 1 ... 1 ... P= 0001. ..00 (4A) 1 1 produces a simple change in the field. The matrix P is a special case . 27Tk x of circulant matrices. The eigenvalues of P are exp. ( — ) and the eigenvectors are 27TR . 27Tk(N-2) . 27Tk(N-l) -1/2. J N~ J N J N . A k = N (1, e , , e , e ) (5A) From these eigenvectors we can obtain the basis of our vector space. The eigenvectors of P for the case N = 4 are A 1 = 1/2 (1, j, -1, -j) A 2 = 1/2 (1, -1 ; 1, -1) A 3 = 1/2 (1, -j- -1, j) (6A) A 4 = 1/2 (1, 1, 1, 1) Note that A A and A satisfy the condition of Equation (1A) whereas A does not. The former three vectors provide an orthonormal basis which spans the vector space of all possible excitations when N = 4. 28 A . A * = 1/4 (1 + 1 + 1 + 1) = 1 A 2 ' A 2 = 1/4 (1 + 1 + 1 + 1) = 1 A 3 . A * = 1/4 (1 + 1 + 1 + 1) = 1 A . A * = 1/4 (1 - j - 1 + j) « (7A) A . A * = 1/4 (1 - 1 + 1 - 1) = A 2 . A 3 * = 1/4 (1 - j - 1 + j) = Let us now examine the properties of these basic excitations. Excitation I = A produces field in which a rotation by 77/2 is equivalent to a 7T/2 change in phase. Solutions of Maxwell's equations in spherical coordinates can be written with azimuthal variation of the form e (m integer) Of these solutions only those with m = 4k + 1 (k = 0, + 1, + 2, +3...) will satisfy the above relationship between rotation and change in phase. Hence the excitation I = A will produce fields which can be expressed in the following way F, = L a (P k 4k + 1 (8A) where <Pm = g (r 9) e J in ^ nfP Similarly, I = A yields m = 4 k + 2 and I = A yields m = 4 k + 3, ^ 3 and these each will produce fields F = E a <P P k 4k + p (9A) A Bimil.ir ftrgtUMnl Cftl i <«d through for arbitrary N. Ari satisfying (1A) can be expressed as a linear combination A ] " A 2' A 3 '' Produced thereby as a linear combination of F ji y y T, " 15i '><>>> de«( ribed by E will produce the fields 29 F = E (A * E*) F (10A) J J If the coefficients a above were known, the fields produced by any excitation would be completely determined. Unf ortunately, this problem has not yet been solved for log-spiral elements. Some useful observations can be made for the log-spiral however, by interpreting the above results in terms of experimental data. Consider the case N = 2 which has been extensively investigated. The eigenvectors for the permutation matrix in this case are \ " \ (1 > ^ A 2 = 1(1,D (11A) and only A satisfies Equation (1A) . This corresponds to the excitation of the two-arm spiral sketched in Figure 3(a) . The field F for the case N = 2 would be of the form F = £ a (P Ik T 2k + 1 (12A) and the possible ^-variations are e j(2k + 1) 9 It has been observed that the radiation produced by 2-arm log-spiral antennas is very nearly circularly-polarized over the major portion of the beam. Circular polarization with non-zero field on the axis requires that E^ and Eg vary as cos(<»>t + <f) and sin (<*>t + <P) respectively, the m = 1 case. Functions with higher values of m would contribute to the off-axis fields. Figure 2A shows a sketch of how the circularly polarized fields = 0, m=2 m = 3 7.h Instantaneous electric vectors at ■ and ■ for three values of m. 31 would appear instantaneously around the polar axis for a few of the small values of m. The values m = 1 and 3 would predominate for the two-arm case. Note that these add alternately in and out of phase in orthogonal cross-sections. Hence the rotational symmetry of the beam should provide an indication of the relative magnitude of these two fields, neglecting all others. A tightly- wound spiral produces more nearly rotationally-symmetric beams, hence predominantly the m = 1 case. The excitation of Figure 3(c) used to produce the conical beams is one of the eigenvectors of (6A), namely A . This excitation produces fields F = £ a, <P 2 k 4k + 2 and the experimental results indicate that, for certain parameter choices, the lowest order terms of this series predominate F ~ a <P n + a <P „ 2 ~ o 2 -1-2 These two cases seem equally likely on the basis of the order of the functions involved. However these two cases correspond to circular polarized fields of opposite sense. One sense of polarization is favored over the other by the direction of the spiral winding. Hence it is reasonable that one of the coefficients a . a be small, a conjecture which is again o' -1 confirmed experimentally.